Maximum Hitting for n Sufficiently Large

Abstract

For a left-compressed intersecting family A ⊆ [n] (r) and a set X ⊆ [n], let A(X) = {A ∈ A : A ∩ X = ∅}. Borg asked: for which X is |A(X)| maximised by taking A to be all r-sets containing the element 1? We determine exactly which X have this property, for n sufficiently large depending on r.Question 2. For which X do we have |A(X)| ≤ |S(X)| for all left-compressed intersecting families A?Borg asked this question in [2], giving a complete answer for the case |X| ≥ r and a partial answer for the case |X| < r. Call X good (for n and r) if for every left-compressed intersecting family A ⊆ [n] (r) we have |A(X)| ≤ |S(X)|.

Classification of Maximum Hittings by Large Families

“…Observe that since every element of A is a potential generator of A, the canonical generating family G of A is indeed a generating family of A. Also note that by definition G must be an antichain, meaning that no element of G is a proper subset of another element of G. Our next proposition establishes the key property that G is supported on the first 2r elements of [n], and is in fact essentially unique in having this property (the existence of a generating family with this property can also be obtained from results of Barber [2]; see Lemma 8 and the discussion preceding it). Lemma 7.…”

EKR Sets for Large n and r

“…In [3], Borg classified X that are EKR for |X| ≥ r and gave a partial solution in the case |X| < r. Barber continued with Borg's work in [2] by considering |X| ≤ r. To describe his results, we introduce the notion of eventually EKR sets, which are sets X ⊂ [2, n] such that for fixed r, we have that X is EKR for sufficiently large n. Barber asked which n are sufficiently large to imply X is EKR. This paper provides bounds on n.…”

“…Based on numerical results for small n and r stated in [2], Barber speculated that n ≥ 2r + 2 was sufficient to imply X is EKR. However, as will be seen in Section 2, this bound does not hold in general.…”

“…Then n > ϕ 2 r implies X is EKR, where ϕ = 1+ √ 5 2 . Note that using the discussion after Theorem 4 in [2], it is easy to show the conjecture holds in the case r = 2, so we address the case r ≥ 3.…”

“…A maximal compressed, intersecting family A is one that is not properly contained in a compressed, intersecting family. Barber proved in [2] that every maximal compressed, intersecting A can be expressed as F (r, n, G) for some G. Thus to show that a given X is EKR, it suffices to show |A(X)| ≤ |S(X)| for intersecting families A of the form A = F (r, n, G). Observe that such families are naturally compressed.…”